All Questions
Tagged with geometric-group-theory mapping-class-groups
30 questions
18
votes
2
answers
790
views
The kernel of the map from the handlebody group to Outer automorphisms of a free group
Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
- 6,229
16
votes
1
answer
1k
views
Mapping class group and CAT(0) spaces
I hope the questions are not too vague.
Is the mapping class group of an orientable punctured surface $CAT(0)$ ?
Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
- 828
14
votes
2
answers
609
views
Symplectic mapping class group and the "Lagrangian sphere complex"
For a genus $g$ surface $\Sigma_g$, the mapping class group $\mathrm{Mod}(\Sigma_g)$ acts on the curve complex $\mathcal C(\Sigma_g)$: vertices being isotopy classes of essential, nonseparating, ...
- 765
13
votes
2
answers
697
views
Mapping Out(F_n) to the mapping class group
Let $\mathrm{Out}(F_g)$ denote the automorphism group of a free group, and $\mathrm{Mod}_g$ the mapping class group of a closed oriented genus $g$ surface. Is there a map, as indicated with the dashed ...
- 40.2k
8
votes
2
answers
566
views
Pseudo-Anosov maps with same dilatation.
Let $S$ be a hyperbolic surface. Suppose $\mathcal{T}$ denotes the Teichmuller space of $S$ and $Mod(S)$ denotes the mapping class group of $S$. Given any pseudo-Anosov element $f\in Mod(S)$, suppose $...
- 1,713
8
votes
2
answers
695
views
Braid groups and Kazhdan's property (T)
In Nica's dissertation Group actions on median spaces, we can read the following assertion:
Braid groups do not contain infinite subgroups satisfying Kazhdan's property (T).
This is used in order to ...
- 8,401
8
votes
1
answer
721
views
Surface groups and non separating loops
QUESTION: Let $g \geq 4$, $S(g)$ be the fundamental group of the genus $g$ surface, and $G$ be finitely generated (the number of generators $\leq 3$) group with abelianization of rank less than equal ...
- 82
8
votes
1
answer
349
views
Subgroups of the mapping class group of a surface generated by Dehn twists
Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a ...
- 113
7
votes
2
answers
353
views
Finite normal subgroup of mapping class group
Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\...
- 605
6
votes
2
answers
522
views
generators for the handlebody group of genus two
Is the handlebody group of genus two surface generated by Dehn twists along properly embedded disks and annuli?
Are there alternative ways to describe a set of generators that are conceptually simple ...
- 858
6
votes
1
answer
658
views
Generalized Birman exact sequence for surfaces with boundaries
Let $S_g^n$ be a surface of genus g with n boundaries and let $Mod(S_g^n)$ be its mapping class group.
We will also denote by $S_{g,m}^n$ a surface of genus g with n boundaries and m punctures.
The ...
6
votes
1
answer
257
views
Explicit description (=pictures!) of elements in $Mod_g[k]$?
This question is probably obvious to experts but I couldn't find the answer in the literature...
Background: Consider the mapping class group $Mod_g$ of the closed genus $g$ surface. There are many ...
- 1,981
6
votes
1
answer
166
views
Translation length on annular curve graphs
Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked.
Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...
- 312
5
votes
2
answers
530
views
Finiteness properties of mapping class groups
Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses:
1) Does every finite generating set give us a finite ...
- 858
5
votes
1
answer
787
views
The action of the mapping class group of a punctured disk on the boundary at infinity of the universal cover
Let $\mathbb{D}\subset\mathbb{C}$ be the unit disk, and remove $n\geq 2$ of its points $P$. The resulting object will be called the punctured disk $\mathbb{D}_n$ in the following. I am interested in ...
- 153
5
votes
0
answers
228
views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
5
votes
0
answers
157
views
Relation between point pushing pseudo-Anosov map and the minimum length
Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space.
By Birman exact sequence we get the point pushing map $Push:\pi_1(S,...
- 1,713
4
votes
2
answers
330
views
Nielsen-Thurston decomposition from the product of Dehn twists
Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is
Given an element as a product of Dehn twist, is it possible ...
- 1,713
4
votes
2
answers
666
views
Action of Mapping Class Group on Arc complex
Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...
- 1,713
4
votes
1
answer
236
views
Mapping class groups of Haken Seifert 3-manifolds (not small)
This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it.
I want to ...
- 309
4
votes
0
answers
209
views
Image of the mapping class group of surfaces into automorphism group?
Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
- 347
3
votes
1
answer
527
views
How do Dehn functions of special linear and mapping class groups behave?
Hi,
I apologize for the basic questions. I am looking for good references on the following problems:
1) What is known about the Dehn function of $SL_n(\mathbb{Z})$?
2) What is known about the Dehn ...
- 33
3
votes
0
answers
228
views
What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?
Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
- 181
3
votes
0
answers
141
views
Translation discreteness of pseudo-Anosov elements
Consider the action of mapping class groups on the curve graphs. A pseudo-Anosov element $g$ acts by a loxodromic isometry on the hyperbolic curve graphs. The translation length of $g$ is the limit $\...
- 619
3
votes
0
answers
414
views
Geometric intersection number for product of elements of the fundamental group
Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...
- 1,713
2
votes
1
answer
245
views
Length of a simple closed curve under Pseudo-Anosov maps
Let $S$ be a fixed hyperbolic surface with genus $g$ and $n$ punctures. Given any pseudo-Anosov map $f$ on $S$ (with stretch factor $\lambda$) with stable and unstable measured foliations $\mu^s$ and $...
- 1,713
2
votes
1
answer
293
views
Quasi-isometric embeddings of the mapping class group into the Teichmuller space
Does there exist a quasi-isometric embedding
$$MCG(S) \to (\mathrm{Teich}(S), d)$$
for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?
- 828
2
votes
1
answer
97
views
Fundamental domain for the action on curve complex
Suppose $S$ is a surface and the Mapping Class Group, $Mod(S)$, of $S$, is the group of self-homeomorphisms of $S$, up to isotopy. This group acts on a graph called "curve graph", denote by $...
- 301
2
votes
0
answers
139
views
Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?
As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.
There are a relationship between the mapping class ...
- 347
1
vote
0
answers
153
views
Topological entropy and pseudo-Anosov dilatation for punctured surface
Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ punctures. Assume $2-2g-n<0$. Let $f$ be a pseudo-Anosov mapping class with dilatation $\lambda_f$. In the introduction (1st page) of the ...
- 1,713